The readers are referred to Adhikari and Bhattacharya (2010), Bhattacharya and Adhikari (2011), and Arany et al. (2015a,b, 2016, 2017) for comprehensive understanding. The fundamentals are provided here.

The equation of motion using the Euler‐Bernoulli beam model for a beam with an axial force is

B.1

where EI(z) is the bending stiffness distribution along the axial coordinate z, μ(z) is the distribution of mass per unit length, P^* is the axial force acting on the beam due to the top head mass and the self‐weight of the tower, p(z, t) is the excitation of the beam, w(z, t) is the deflection profile.

Using constant equivalent values for the axial force, bending stiffness and mass per length, and considering free harmonic vibration of the beam with separation of variables w(z, t) = W(z) · e^iωt, the equation can be reduced to the following using the non‐dimensional parameters of Table B.1 and the dimensionless axial coordinate ξ = z/L:

B.2

Dimensionless group	Formula	Dimensionless group	Formula
Nondimensional lateral stiffness		Nondimensional axial force
Nondimensional rotational stiffness		Mass ratio
Nondimensional cross stiffness		Nondimensional rotary inertia

K_L, K_R, K_LR are the lateral, rotational, and cross stiffness of the foundation, respectively; EI_η is the equivalent bending stiffness of the tapered tower; L_T is the hub height above the bottom of the tower; P^* is the modified axial force, m_RNA is the mass of the rotor‐nacelle assembly; m_T is the mass of the tower; J is the rotary inertia of the top mass; μ is the equivalent mass per unit length of the tower.

where is the nondimensional axial force and is the non‐dimensional circular frequency).

Using the nondimensional numbers as defined above, the boundary conditions can be written for the bottom of the tower (ξ = 0):

B.3

B.4

and the top of the tower (ξ = 1):

B.5

B.6

The parameters used in the boundary conditions are defined in Table B.1. The characteristic equation for the equation of motion can be written as

B.7

and

The four solutions are then

B.8

with which the solution is in the form

B.9

which can be transformed using Euler's identity to

B.10

with

B.11

Substituting this form of the solution into the boundary conditions, one obtains four equations, written in matrix form as

B.12

with

B.13

and

B.14

Looking for nontrivial solutions of this equation one obtains

B.15

from which one can obtain the nondimensional circular frequency Ω, and from that the natural frequency using

B.16

The equation that has to be solved is transcendental and therefore solutions can only be obtained numerically.